Let p and q be integers and define r = pq + p + q. Prove that r is even if and only if p and q are both even.

The value of r can only be even if both value of P and q are even

Exploring the 3 possible scenarios:

If p is even and q is odd:

the value of Pq is even and P ₊ q is odd

Hence, the value of r (r = pq + p + q) is : even number ₊ odd number = odd number

e. g if p=2 and q=3

r = pq + p + q = 2*3 + 2+3 = 6 + 5 = 11

and If p is odd and q is odd:

the value of Pq is odd and P ₊ q is even

Hence, the value of r (r = pq + p + q) is : odd number ₊ even number = odd number

e. g if p=5 and q=3

r = pq + p + q = 5*3 + 5+3 = 15 + 8 = 23

If p is even and q is even:

the value of Pq is even and P ₊ q is even

Hence, the value of r (r = pq + p + q) is : even number ₊ even number = even number

e. g if p=2 and q=4

r = pq + p + q = 2*4 + 2+4 = 8 + 6 = 14